Introduction to logic a proposition is a statement

Introduction to LogicApropositionis a statement that is either true or false.For example, “2 + 2 = 4” and“Donald Knuth is a faculty at Rutgers-Camden” are propositions, whereas “What time isit?”,x2< x+ 40 are not propositions.We can construct compound propositions from simpler propositions by using some of thefollowing connectives. Letpandqbe arbitrary propositions.Negation:˜p(read as “notp”) is the proposition that is true whenpis false and vice-versa.Conjunction:p∧q(read as “pandq”) is the proposition that is true when bothpandqare true.Disjunction:p∨q(read as “porq”) is the proposition that is true when at least one ofporqis true.Exclusive Or:p⊕q(read as “pexclusive-orq”) is the proposition that is true whenexactly one ofpandqis true is false otherwise.Implication:p→q(read as “pimpliesq”) is the proposition that is false whenpis trueandqis false and is true otherwise.

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6Lecture OutlineJanuary 10, 2011The implicationq→pis called theconverseof the implicationp→q. The implication¬p→ ¬qis called theinverseofp→q. The implication¬q→ ¬pis thecontrapositiveofp→q.ponly ifqmeans “if not q then not p”, or equivalently ifpthenq.Biconditional:p↔q(read as “pif, and only if,q”) is the proposition that is true ifpandqhave the same truth values and is false otherwise. “If and only if” is often abbreviated as iff.The following truth table makes the above definitions precise.pq¬pp∧qp∨qp⊕qp→qq→pp↔qTTFTTFTTTTFFFTTFTFFTTFTTTFFFFTFFFTTTNecessary and Sufficient Conditions:For propositionspandq,pis asufficientcondition forqmeans thatp→q.pis anecessarycondition forqmeans that¬p→ ¬q, or equivalentlyq→p.Thuspis a necessary and sufficient condition forqmeans “piffq”.Logical EquivalenceTwo compound propositions are logically equivalent if they always have the same truthvalue. Two statementpandqcan be proved to be logically equivalent either with the aidof truth tables or using a sequence of previously derived logically equivalent statements.Example.Show thatp→q≡ ¬p∨q≡ ¬q→ ¬p.Solution.The truth table below proves the above equivalence.pq¬p¬qp→q¬p∨q¬q→ ¬pTTFFTTTTFFTFFFFTTFTTTFFTTTTTExample.Show thatp≡ ¬p→C.p¬pC¬p→CTFFTFTFFThe above equivalence forms the basis of proofs by contradiction.

January 10, 2011Lecture Outline7The logic of Quantified StatementsConsider the statementx <15. We can denote such a statement byP(x), wherePdenotesthe predicate “is less than 15” andxis the variable.This statementP(x) becomes aproposition whenxis assigned a value. In the above example,P(8) is true whileP(18) isfalse.

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